## Dissertations from ProQuest

#### Title

Caustic Aided Parameter Estimation: An Application To Geophysical Travel Time Inversion

1987

Article

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Applied Marine Physics/Ocean Engineering

#### Abstract

A processing scheme has been developed that offers a solution to the long standing problem of temporally resolving the overlapping, narrowband arrivals associated with a geophysical travel time triplication. The scheme, which we refer to as caustic aided parameter estimation (CAPE), is based on the observation that a travel time triplication is due to the presence of a cusp caustic. Ideas from the application of catastrophe theory to caustic structures have been embodied in a two parameter time delay model and then assembled as a bank of correlation detectors. A seismogram is passed through this bank of detectors to yield estimates for the three arrival times. A particularly interesting result is that the trajectory of the estimates in parameter space appears to be a robust indicator of the presence of a triplication.We first develop the processing scheme in the context of a set of synthetic WKBJ seismograms containing a triplication. Monte Carlo simulations indicate time resolutions for a 10 Hz bandwidth source on the order of $\pm$10 ms for a 20dB SNR and $\pm$15 ms for a 10dB SNR. From the triplication estimate obtained from the CAPE processor, we use an extremal method to estimate X(p) and $\tau(\rm p)$ for use in the Herglotz-Wiechert inversion formula. We have found that the inversion of $\tau(p)$ gives better results than the inversion of X(p). Monte Carlo simulations for $\tau(p)$ inversions indicate depth error bounds on the order of $\pm$0.02 km for a 20dB SNR and $\pm$0.05 km for a 10dB SNR. In both cases, the maximum bias error was on the order of 0.05 km. Simulations carried out for a variable number of sensors indicate that the number of sensors should be maximized. We have further developed a constrained version of the CAPE methodology and applied it to a measured marine refraction data set (Fanfare 2 P-wave data) and resolved and inverted a triplication associated with the Moho. The top of the Moho was found to be at 9.54 km with a velocity of 7.25 km/s and the bottom at 9.83 km with a velocity of 7.66 km/s.

Geophysics